process

344 U. Bunke, T. Schick, M. Spitzweck, and A. Thomby the diagramBT jBCBT jB B BT jBP ˝EzG P ZjBzG P B zG(72)Z jBdiagZ jB B Z jB .The right lower square is cartesian, and in the left upper square we take the fibre-wisequotient by the anti-diagonal action of BT jB .6.4.13 We have D.P / 2 Ext PIC.S=B/ .Z;D.E//, where0 ! BT jB ! D.E/ ! D.E/ ! 0:We define D.P / to be the quotient of D.P / by BT jB in the sense of 2.5.12 so thatD.P / ! D.P / is a gerbe with band T.We define D.P / ˝D.P /D. zG/ by the diagramBT jBCBT jB B BT jBD.P / ˝D.P /D. zG/ D.P / ZjB D. zG/ D.P / B D. zG/(73)Z jBdiagZ jB B Z jB .Again, the right lower square is cartesian, and in the left upper square we take thefibre-wise quotient by the anti-diagonal action of BT jB .Proposition 6.22. We have an equivalence of Picard stacksD.P ˝EzG/ Š D.P / ˝D.P /D. zG/:Proof. The diagram (73) defines D.P / ˝D.P /D. zG/ by forming the pull-back to thediagonal and then taking the quotient of the anti-diagonal BT jB -action in the fibre.One can obtain this diagram by dualizing (72) and interchanging the order of pull-backand quotient.